Optimization and Bounded Constraints

A small workshop produces two items, Item X and Item Y. Each unit of Item X requires $2$ hours of labor and yields a profit of $\$30$. Each unit of Item Y requires $4$ hours of labor and yields a profit of $\$50$. The workshop has a maximum of $16$ labor hours available daily and must produce a combined total of at least $5$ units per day. Assuming only whole units can be produced, which combination of (Item X, Item Y) maximizes total daily profit while satisfying all given constraints?

A factory manufactures two products, Product X and Product Y. Daily production is subject to the following constraints:
- Each unit of Product X requires $1$ hour of machine time.
- Each unit of Product Y requires $2$ hours of machine time.
- Total machine time available per day is at most $10$ hours.
- The factory must produce at least $2$ units of Product X and at least $2$ units of Product Y per day.
- Each unit of Product X yields a profit of $\$15$, and each unit of Product Y yields a profit of $\$20$.

Match each optimization metric on the left with its corresponding correct numerical value on the right.

A logistics company operates two types of delivery vehicles: Type X and Type Y.

- Each Type X vehicle carries $40$ packages and costs $\$300$ per trip.
- Each Type Y vehicle carries $25$ packages and costs $\$150$ per trip.

The company operates under a maximum daily budget of $\$3,300$. Additionally, due to fleet maintenance regulations, the number of Type Y vehicle trips cannot exceed twice the number of Type X vehicle trips.

Which combination of Type X and Type Y vehicle trips maximizes the total package delivery capacity while satisfying all daily budget and maintenance constraints?

An analytics firm processes data batches using two types of cloud computing instances: High-Memory Instances ($M$) and High-Compute Instances ($C$).

- Processing Capacity: Each High-Memory Instance processes $120$ thousand transactions per hour, and each High-Compute Instance processes $200$ thousand transactions per hour. The workload requires a total processing rate of at least $1,320$ thousand transactions per hour.
- Instance Availability: At most $10$ High-Memory Instances ($M \le 10$) and at most $6$ High-Compute Instances ($C \le 6$) are available. At least $1$ instance of each type must be used ($M \ge 1$ and $C \ge 1$).
- Load Balancing Constraint: To ensure infrastructure stability, the number of High-Memory Instances cannot exceed twice the number of High-Compute Instances ($M \le 2C$).
- Operating Costs: High-Memory Instances cost $\$18$ per hour each, while High-Compute Instances cost $\$25$ per hour each.

Select the number of High-Memory Instances ($M$) and the number of High-Compute Instances ($C$) that minimize the total hourly operating cost while satisfying all operational requirements.

A commercial greenhouse allocates integer numbers of acres to grow two high-yield crops: Organic Tomatoes ($T$) and Hydroponic Cucumbers ($C$). The operational parameters and constraints are as follows:

- Each acre of Tomatoes requires $3$ units of water per day and $4$ units of fertilizer per week, producing a net revenue of $\$1,200$ per week.
- Each acre of Cucumbers requires $5$ units of water per day and $2$ units of fertilizer per week, producing a net revenue of $\$1,000$ per week.
- Total daily water usage across both crops cannot exceed $45$ units.
- Total weekly fertilizer usage across both crops cannot exceed $36$ units.
- To fulfill distributor agreements, the total cultivated area ($T + C$) must be at least $10$ acres.

Which of the following crop allocation pairs $(T, C)$ maximizes total weekly net revenue while satisfying all operational constraints?

A research laboratory is formulating a daily dosage protocol for a clinical trial combining two therapeutics, Drug A ($A$) and Drug B ($B$), measured in integer milligrams (mg).

The trial protocol specifies the following operational constraints:
- The daily dosage of Drug A must be at least $10\text{ mg}$ and at most $40\text{ mg}$ ($10 \leq A \leq 40$).
- The daily dosage of Drug B must be at least $15\text{ mg}$ and at most $50\text{ mg}$ ($15 \leq B \leq 50$).
- To prevent hepatotoxicity, the combined daily dosage $(A + B)$ cannot exceed $65\text{ mg}$ ($A + B \leq 65$).
- To ensure therapeutic efficacy, the dosage of Drug B must be at least $10\text{ mg}$ less than twice the dosage of Drug A ($B \geq 2A - 10$).

The total treatment efficacy score $E$ is modeled by the linear function $E = 3A + 4B$.

Based on the constraints above, select the daily dosage for Drug A and the daily dosage for Drug B that together maximize the total treatment efficacy score $E$.

A pharmaceutical laboratory synthesizes two custom therapeutic compounds, Compound $X$ and Compound $Y$, in integer batch quantities during a single production cycle. Production is subject to the following resource and operational constraints:

- Bioreactor Time: Each batch of Compound $X$ requires $4$ hours and each batch of Compound $Y$ requires $3$ hours. The laboratory has at most $48$ total hours of bioreactor time available.
- Specialized Solvent: Each batch of Compound $X$ requires $3$ liters and each batch of Compound $Y$ requires $5$ liters. The total solvent supply is capped at $55$ liters.
- Catalyst Stability Limit: To prevent reactive degradation, the number of batches of Compound $Y$ produced cannot exceed twice the number of batches of Compound $X$ plus $1$ ($Y \le 2X + 1$).

Each batch of Compound $X$ generates a net profit of $\$12,000$, and each batch of Compound $Y$ generates a net profit of $\$14,000$.

What is the maximum total net profit, in dollars, that the laboratory can achieve within these combined production constraints?

An earth-observation satellite payload operator allocates integer numbers of channels to two operational payload modes: High-Resolution Imaging ($I$) and Atmospheric Sounding ($S$). Operational parameters and resource bounds are specified as follows:

- Bandwidth: Each imaging channel requires $15\text{ MHz}$ and each sounding channel requires $10\text{ MHz}$. Total available payload bandwidth is at most $210\text{ MHz}$.
- Power: Each imaging channel consumes $40\text{ W}$ and each sounding channel consumes $15\text{ W}$. Total available power budget is at most $440\text{ W}$.
- Mission Minimums: The satellite must operate at least $2$ imaging channels ($I \ge 2$) and at least $4$ sounding channels ($S \ge 4$).
- Buffer Constraint: The number of sounding channels cannot exceed twice the number of imaging channels ($S \le 2I$).

Match each payload optimization target on the left with its corresponding integer channel count on the right.

A regional logistics operator manages cargo transit on a river corridor using two types of vessels: Express Barges ($E$) and Heavy-Haul Barges ($H$). Weekly operational constraints are defined as follows:

- Fuel Allowance: Each Express Barge consumes $8$ metric tons of fuel per trip, and each Heavy-Haul Barge consumes $12$ metric tons. Total weekly fuel consumption cannot exceed $144$ metric tons.
- Crew Availability: Each Express Barge requires $3$ crew shifts, and each Heavy-Haul Barge requires $2$ crew shifts. Total available crew shifts per week cannot exceed $30$.
- Service Minimum: The operator must deploy at least $2$ Heavy-Haul Barges ($H \ge 2$) per week to maintain baseline commercial obligations.
- Profit Structure: Each Express Barge generates a net profit of $\$5,000$, and each Heavy-Haul Barge generates a net profit of $\$6,000$.

Match each operational metric on the left with its correct optimal value under the profit-maximizing schedule of integer barge deployments.

A cloud analytics company configures daily data processing operations using two types of virtual server instances: Compute-Optimized ($C$) and Memory-Optimized ($M$). The operational parameters and constraints are as follows:

- Each $C$ instance processes 50 batch jobs per hour and incurs an operating cost of $15 per hour. - Each$ M $instance processes 30 batch jobs per hour and incurs an operating cost of$ 10 per hour.
- The system must process at least 360 batch jobs per hour in total.
- The total hourly operating budget for server instances cannot exceed $115. - Due to active database caching policies, at least 3$ M $instances must be active at all times ($ M \ge 3 $). -$ C $and$ M$ must both be non-negative integers.